anonymous
  • anonymous
I need help understanding integration by substitution: 63/(9x+2)^8
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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phi
  • phi
if you let u = 9x+2 what is du ?
xapproachesinfinity
  • xapproachesinfinity
This is the as saying f(x) =9x+2 What is f'(x) Just a difference of differentials but you don't need to worry about thst
anonymous
  • anonymous
du=9 right?

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xapproachesinfinity
  • xapproachesinfinity
Yes just one thing you missed Du=9dx
phi
  • phi
no du/dx =9 but if you "take the derivative" u= 9x you differentiate the variables, in this case the u and the x like this du = 9 dx
phi
  • phi
and once you have du = 9 dx then you also have \[ dx = \frac{1}{9} du \]
phi
  • phi
now do your variable substitution replace 9x+2 with u replace dx with 1/9 du
phi
  • phi
and if we have limits, we would also change the limits.
phi
  • phi
and because it is easy to integrate a power, I would use the -8 exponent and get rid of the fraction
anonymous
  • anonymous
So would it be: \[63u ^{-8}*1/9 du\] ??
phi
  • phi
yes and the problem is \[ 7 \int u^{-8} du \] which you can do , right?
xapproachesinfinity
  • xapproachesinfinity
Yes
phi
  • phi
once you integrate, replace u with 9x+2 to put the answer in terms of x
SolomonZelman
  • SolomonZelman
\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{63}{(9x+2)^8}~dx}\) \(\large\color{slate}{\displaystyle u=9x+2}\) \(\large\color{slate}{\displaystyle du=(9x+2)'~\cdot dx~~~\rightarrow~~~du=9~dx}\) (you already have a 9 and dx to replace that by du, just need to re-write it. \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7\cdot 9}{(9x+2)^8}~dx}\) \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(9x+2)^8}~(9\cdot dx)}\) substitution:: \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(u)^8}~(du)}\) and then all you have left:: \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}7~u^{-8}~du}\)
SolomonZelman
  • SolomonZelman
(APPLY THE POWER RULE, AND don't forget to substitute the x back for u.)
SolomonZelman
  • SolomonZelman
igtg....
anonymous
  • anonymous
Ok, I got it! Thank you all!

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